Chemistry 3820 Lecture Notes Dr. M. Gerken Page62
6.2. Introduction to Spectroscopic states and term symbols
From the number of absorption bands we have already seen that usually more d-d transitions are observed than expected
from a simplified picture where one electron is excited from a t2g orbital to an eg orbital. An electron configuration, e.g.,
(t2g)1(eg)1, turns out to be a rather simplistic picture. For a better, more detailed description, spectroscopists introduced term
symbols. When labeling spectroscopic states for transition-metal complexes, we have two choices to describe these states. (a) We
can look at the isolated (gas-phase) metal cation, start from its electron configuration (dn), and look at the ground and excited
states. (b) On the other hand, we can look at the complex of a certain point symmetry and use appropriate Mulliken labels.
6.2.1 Gas-phase metal cation:
A single electrons exhibit two types of angular momenta: a spin (intrinsic angular momentum) and an orbital angular
momentum. The quantum numbers that describe the size of the angular momenta are: s (for the spin angular momentum, which is
always ½ for a single spin) and l (for the orbital angular momentum). The quantum numbers, ms and ml, specify the orientation of
the angular momenta with respect to a primary axis (the z-axis). As discussed in an earlier chapter, ml (ms) can have values of l,l-
1, l-2,…-l (s, s-1, s-2,…., -s; only ½ and – ½ are possible, due to the restriction of s). Another way of looking at these quantum
numbers is the consideration that l and s are given my the largest number of ml and ms.
In order to describe a spectroscopic state of a cation, we have to consider the total angular momentum L and the overall
electron spin S. Both are related to the angular momentum quantum number, l, and spin quantum number, s, for single electrons,
respectively. We also have to consider Ml and Ms quantum numbers for the atoms/cations.
(i) Term symbols for the Mg atom:
Let’s start with a simple example: the neutral Mg atom. In the ground state, both valence electrons are in an s orbital with
opposite spin directions:
electron(1): ml=0, ms = ½
electron(2): ml=0, ms= -½
Because of the Pauli exclusion principle, we do not have another choice in combining these electrons differently in this ground
state. The whole atom in the ground state has a total angular momentum: ML = ml(1) + ml (2) = 0+0 =0
and an overall spin: Ms = ms+ ms =½+(-½) =0.
Therefore, our L and S values for the Mg are both 0. Instead of using these two numbers directly for the description of the state,
spectroscopists use term symbols. Similar to the orbital notation (s orbital for l=0, p orbital for l=1, etc), the following
convention is used: L= 0 1 2 3 4 ….
S P D F G then alphabetical
Note that capital letters are used for term symbols. The spin is described by the spin multiplicity = 2S+1 and placed as the
superscript in front of the angular momentum notation. A multiplicity, 2Q+1, provides information about how many different
states are possible with a particular quantum number, Q. We can also define an orbital multiplicity, 2L+1. For example, a P term
has an orbital multiplicity of 2·1+1 = 3; three states are described by P. Compare this to the number of p orbitals. The ground
state of the Mg atom is therefore described with the term 1S (read: singlet S).
In the excited state of Mg*, one electron is excited to a p orbital. Now we have to distinguish between different microstates,
resulting in 12 possible combinations. Microstates can be best illustrated by placing electrons in orbitals, which are defined by the
quantum numbers, ml (and ms). The spins of the s- and p electrons can be parallel or antiparallel and the excited electron can
reside in the px, py, or pz orbital. In the absence of electron interaction (repulsion) all microstates would be degenerate. Due to
electron repulsion, the degeneracy between the microstates is lifted and we can obtain the following terms symbols, which
describe groups of microstates. Let’s start with the distinction between the parallel and antiparallel spin
Parallel maximum ML = l(1) + l(2) = 0+1 =1 maximum MS = ms+ ms =½+½ =1
Therefore, L = 1 and S =1 term symbol: 3P (read: triplet P)
Antiparallel maximum ML = l(1) + l(2) = 0+1 =1 maximum MS = ms+ ms =½+(-½) =0
Therefore, L = 1 and S =0 term symbol: 1P (read: singlet P)
The term symbol P describes three degenerate microstates: spin multiplicity of 1 and orbital multiplicity of 3. On the other hand,
the 3P term describes nine degenerate microstates, arising from a spin multiplicity of 3 and an orbital multiplicity of 3. This way
we accounted for all microstates.
(ii) Term symbols for a d2 configuration: Ti2+
The first step is to write down all microstates for the ground state of Ti2+. In Table 13.1 of the text book (p. 441) all microstates
are given for the d2 configuration in terms of (ml(1)+/-,ml(2) +/-) - + and – indicating a ms values for + ½ and – ½. These
microstates are represented by the terms, 1G, 3F, 1D, 3P, and 1S. Taking into account the spin and orbital multiplicities of each
term, we obtain 45 microstates: Term Number of microstates
G 9×1= 9
F 7×3= 21
D 5×1= 5
P 3×3= 9
S 1×1= 1
Chemistry 3820 Lecture Notes Dr. M. Gerken Page63
(ii) Relative energies of spectroscopic terms
We have encountered Hund’s rule as a rule how to fill orbitals with electrons. In order to distinguish the relative energies of
terms, and especially to identify the ground state, we can use the extended version of Hund’s rules.
(1) The term with the greatest spin multiplicity lies lowest in energy.
This rule states that the triplet states are more stable than singlet states. This is in accordance with the simpler version of Hund’s
rule, that states that electrons in different orbitals with the same spin direction are more stable than those with opposite spin
(2) For a given multiplicity, the greater the value of L of a term, the lower the energy.
When L is high, each electron can stay clear of its partner and hence feels lower coulombic interactions.
Using these rules, the following order of terms for Ti2+ is predicted:
F < 3P < 1G < 1D < 1S
However, the observed order is:
F < 1D < 3P < 1G < 1S
It turns out that Hund’s rules are reliable only for the determination of the ground state.
In order to assign energies to terms, electronic repulsion integrals would have to be rigorously analysed. Fortunately, the
repulsion integrals can be combined into three parameters: the so-called Racah parameters A, B, and C. These Racah
parameters are sums of repulsion integrals and the energies of the terms are expressed as linear combinations of the Racah
parameters. For the d2 electron configuration, the energies are given as E(1S) = A +14B +7C
E(1G) = A + 4B +2C
E(1D) = A - 3B +2C
E(3P) = A + 7B
E(3F) = A - 8B
All three Racah parameters are positive and their values of the Racah parameters are determined empirically by fitting the
observed energies of the terms to the above
expressions. Since the first parameter, A, is Racah parameters for some d-block ions: B parameters with the value of
present for all terms, it can be disregarded for C/B in parentheses; B is in cm-1
energy comparisons. 1+ 2+ 3+ 4+
In the case that C > 5B the order of the terms is Ti 720(3.7)
close to that predicted by Hund’s rules. If C < V 765(3.9) 860(4.8)
5B high orbital angular momentum becomes Cr 830(4.1) 1030(3.7) 1040(4.1)
more important compared to having a high spin Mn 960(3.5) 1130(3.2)
multiplicity. Fe 1060(4.1)
The Table on the right-hand side gives Racah Co 1120(3.9)
parameters for some d-block ions. Ni 1080(4.5)
Cu 1220(4.0) 1240(3.8)
6.2.2 Metal cation complexes:
As soon as ligands are coordinated to a transition-metal centre, a molecular symmetry is imposed and the d orbital split
according to the ligand field. In these cases, molecular term symbols have to be used.
Molecular term symbols
Since L is not a good quantum number in complexes, which are non-spherical, states are labeled using symmetry species of
the correct point group. The simplest example is a transition-metal centre with d1 configuration, e.g., the Ti3+ cation. This electron
can reside either in the t2g or eg orbital sets. Therefore, the molecular term symbols for the ground state (electron configuration:
t2g1) is 2T2g and for the excited state (electron configuration: eg1) is 2Eg. To remind us, let’s look at the term symbol for gas-phase
Ti3+. Since we have a single unpaired electron, we
are automatically dealing with doublet states (spin The correlation of spectroscopic terms for d electrons in Oh complexes
multiplicity of two; 2· ½ +1). The is located in a Atomic Term Number of States Terms in Oh Symmetry
d-orbital (l=2,1,0,-1,-2) with a maximum angular S 1 A1g
momentum quantum number of l = 2. Hence, the P 3 T1g
term for the ground state is 2D, which has an D 5 T2g + Eg
orbital multiplicity of 2· 2 +1 =5, which F 7 T1g + T2g + A2g
represents the five different d orbitals which the G 9 A1g + Eg + T1g + T2g
electron can reside in. We are not concerned
about term symbols of the excited states since
Chemistry 3820 Lecture Notes Dr. M. Gerken Page64
those states would involve excitation from or to a different type of orbitals. The ground state of the free ion (2D) splits into the
two molecular terms 2T2g and 2Eg upon formation of an octahedral complex. From ligand field theory, we know that the 2T2g state
is lower in energy compared to 2Eg.
But most of the time electron repulsion must be considered and a eg ← t2g transition is not an unambiguous description of a
transition. For example, dz2 ← dxy and dz2 ← dxz transitions fall into this category, but they are not necessarily degenerate if other
electrons are present. Group-theoretical treatment provides us with tables that correlate atomic terms with molecular terms (table
for Oh complexes shown below). In that table we can find the splitting of the atomic D term that we derived above.
We can draw a correlation diagram between atomic terms and molecular terms. For the d1 electron configuration we split the 2D
term into 2T2g and 2Eg. The degree of splitting depends on the strength of the ligand field and has a value of ∆o as shown below.
For the d2 electron configuration of a transition metal cation, two
triplet terms have to be considered: 3F and 3P. Using the table
with correlates the atomic terms to molecular terms, we can
derive four molecular triplet terms: 3A2g, 3T2g, 3T1g. The 3A2g,
T2g, and 3T1g terms correspond to the eg2, eg1t2g1, and t2g2 electron
configurations, respectively. The energies of these terms can be
written in terms of ∆o: E(t2g2, 3T1g) = 2(-2/5 ∆o) = -0.8 ∆o
E(eg1t2g1, 3T2g) = (-2/5+3/5) ∆o = 0.2 ∆o
E(eg2, 3A2g) = 2(3/5 ∆o) = 1.2 ∆o
As a result, the energies relative to that of the lowest term are:
E(t2g2, 3T1g) = 0
E(eg1t2g1, 3T2g) = ∆o
E(eg2, 3A2g) = 2 ∆o
The correlation diagram above illustrates the splittings of the
molecular terms and their relative energies in terms of ∆o. The
relative energies of the atomic terms are given using the Racah
In these diagrams, we can distinguish between two extreme cases:
(a) the weak-field limit and (b) the strong-field limit. The
weak-field limit is found on the left side of the diagram, where
electron-electron repulsion is dominating the energy and terms
can be expressed as linear combinations of Racah parameters.
Labels using atomic term symbols are a good approximation. In
the strong-field limit, the ligand field is dominating the relative
energies. Molecular term symbols have to be used.
Correlation diagrams between energies of atomic and
molecular terms can drawn as so-called Tanabe-Sugano
diagrams for each electron configuration of free ions. The
ground state is the reference point and is along the x-axis. Energy
Chemistry 3820 Lecture Notes Dr. M. Gerken Page65
values in a Tanabe-Sugano diagram are only given relative to the ground state. The simple correlation diagram had multiples of B
(the Racah parameter) on the energy axis to denote the
relative energies of the atomic terms. In the Tanabe-Sugano
diagrams, the energy axis has units of E/B. The x-axis has
units of ∆o/B. Each Tanabe-Sugano diagram is given for
only one specific B/C ratio (the best value). For example, the
Tanabe-Sugano diagram for d3 complexes is given for C =
On the right-hand side, Tanabe-Sugano diagrams for d2
and d3 configurations are given. So far we have encountered
straight lines in correlation diagrams. The curved lines in the
Tanabe-Sugano diagrams for d2 and d3 configurations are a
consequence of the non-crossing rule. Terms of the same
symmetry cannot cross and will ‘repel’ each other (see
Excitations of electrons are transition between electronic
states. Usually, we are concerned about transitions from the
ground state to an excited state. Therefore, in a Tanabe-
Sugano diagram we start from the x-axis (ground state) and
measure the height of an arrow to an excited state, which
provides us with the energy separation. Depending on the
ligand-field splitting, the energy of these transitions will
vary. How many spin-allowed transition do you expect for
octahedral d2- and d3-transition-metal complexes?
The UV-visible spectrum given above is an absorption
spectrum for a d3 complex. The transitions are indicated in term
symbols as well as with the corresponding electron configuration.
The ratio of energies (wavenumbers) of the two spin-allowed d-d
transitions can be used to obtain experimental values for B and
∆o. The ratio of the energies can be found in the Tanabe-Sugano
diagram, providing a specific value for ∆o/B. The Racah
parameter B can be calculated form the E/B value for the low-energy transitions taking the energy (wavenumber) of the transition
as E. Using this Racah parameter, the ligand field splitting, ∆o, can be calculated.
The nephelauxetic effect
So far, we considered the interelectronic repulsion parameters, B and C (the Racah parameters) as fixed. However, the
experimentally determined value for B of the [Cr(NH3)6]3+ complex is 657 cm-1. For the free Cr3+ ion B is 1030 cm-1, giving a
64% reduction of B in going from the free ion to the complex. This finding means that the electron-electron repulsion between d
electrons is reduced in the hexammine complex, relative to the free ion. The reduction of interelectronic repulsion is a
consequence of delocalization of d-electron density onto the ligands. In MO terms, the distance between the electrons in
complexes increases as a result of the formation of larger MO’s upon combination of metal and ligand orbitals. This effect is
called nephelauxetic effect from the Greek words for ‘cloud expanding’. The reduction of B is usually reported in terms of the
nephelauxetic parameter, β:
β = B(complex)/B(free ion)
Smaller β corresponds to larger reduction of interelectronic repulsion. Different ligands result in different nephelauxetic
parameters. Hence, we can write a nephelauxetic series of ligands:
F- > H2O > Cl- > CN- > Br- > N3- > I-
Soft ligands, like the iodo ligand, form more covalent coordinative bonds, resulting in larger delocalization of the metal-d
Chemistry 3820 Lecture Notes Dr. M. Gerken Page66
We have seen in the previous chapter that some transitions are not allowed, for example transitions where the spin of the electron
in changed are spin-forbidden. This does not mean that such a transition will never occur, but that it is less likely and that the
intensity (molar absorption coefficient) of such an absorption band is very low. Whether transitions are allowed or forbidden, and
to what degree they may be forbidden depends on selection rules:
(i) spin selection rule
Electromagnetic radiation usually cannot change the relative orientation of an electron spin. The selection rule states that the
overall spin S of a complex must not change during an electronic transition, hence, ∆S = 0.
This selection rule can be relaxed by spin-orbit coupling, i.e., coupling between spin angular momentum and orbital angular
momentum. However, spin-forbidden transitions remain generally much weaker than spin-allowed transitions. Spin-orbit
coupling is particularly strong for heavy d-metals. In the 3d series, spin-forbidden transitions have εmax = 1 L mol-1 cm-1.
In an octahedral d3 transition-metal complex, the 2Eg ← 4A2g transition is spin-forbidden. It is still visible as a very weak
absoption band in the UV-visible spectrum (see above).
(ii) Laporte selection rule
The Laporte selection rule is particularly applicable for centrosymmetric molecules, i.e., molecules with an inversion centre.
Orbitals and states in such molecules can be described with the symmetry labels g (gerade) and u (ungrerade). Laporte-allowed
transitions are accompanies with a change of parity (g and u); this means, transitions from a g-state to a u-state and vice versa are
Laporte-allowed and transitions from a g- to a g-state and from a u- to a u-state are Laporte-forbidden. All d orbitals are gerade
(have g symmetry). Therefore, d-d transitions, e.g., 4T2g ← 4A2g in an octahedral d3 complex, are Laporte-forbidden.
The Laporte selection rule is relaxed by (a) deviation of
perfect centrosymmetric symmetry due to ligands of less Intensities of spectroscopic bands in 3d complexes
symmetry or chelate ligands and (b) asymmetric vibrations. Band type εmax / L mol-1 cm-1.
While d orbitals are always centrosymmetric by themselves, for a Spin-forbidden <1
non-centrosymmetric ligand field the Laporte selection rule is Laporte-forbidden d-d 20-100
strictly not applicable anymore. As a consequence, Laporte- Larport-allowed d-d ca. 500
forbidden d-d transitions have εmax = 20 to 100 L mol-1 cm-1, Symmetry-allowed (e.g. CT) 1000 – 50 000
while Laporte-allowed d-d transitions have εmax = ca. 500 L mol-1
(iii) Symmetry-allowed transitions
Transitions which are not governed by symmetry restrictions (e.g. Laporte rule) have much larger intensities in the electronic
absorption spectra. Such transitions are usually charge transfer band that give rise to intense colorations.